Chapter 1 Systems of Linear Equations 1.1Introduction to Systems of Linear Equations 1.2Gaussian Elimination and Gauss-Jordan Elimination 1.3Applications of Systems of Linear Equations 1.1 1.2 1.1 Introduction to Systems of Linear Equations A linear equation in n variables: ai : real-number coefficients xi : variables needed to be solved b : real-number constant term a1: leading coefficientx1: leading variable Notes: (1) Linear equations have no products or roots of variables and no variables involved in trigonometric, exponential, orlogarithmic functions (2) Variables appear only to the first power 1 1 2 2 3 3 n na x a x a x a x b + + + + =1 1(h)4x y+ =1.3 Ex 1: Linear or Nonlinear (a) 3 2 7 x y + =1(b)22 x y z t + =1 2 3 4(c)2 10 0 x x x x + + =21 2(d) (sin ) 42x x et =(e)2 xy z + =(f)2 4xe y =1 2 3(g)sin 2 3 0 x x x + =product of variables is the exponent xtrigonometric functionLinearLinearLinearLinearNonlinearNonlinearNonlinearNonlinearnot the first power1.4 A solution of a linear equation in n variables: Solution set: The set of all solutions of a linear equation b x a x a x a x an n= + + + + 3 3 2 2 1 1,1 1s x = ,2 2s x = ,3 3s x = , n ns x =b s a s a s a s an n= + + + + 3 3 2 2 1 1s.t. Usually, the number of solutions of a linear equation is infinite, so we need some methods to represent these solutions. 1.5 If you choose x1 to be the free variable, the parametric representation of the solution set is Ex 2: Parametric representation of a solution set 4 22 1= + x x If you solve for x1 in terms of x2, you obtain By letting (the variable t is called a parameter), you can represent the solution set as The set representation for solutions: { } R t t t e | ) , 2 4 (1 2 24 2(in this form, the variable is free) x x x = t x =21 24 2 , ,is any real number x t x t t = ={ } R s s s e | ) 2 , (21with a solution (2, 1), i.e.,1 , 22 1= = x x1.6 A system of m linear equations in n variables: 11 1 12 2 13 3 1 121 1 22 2 23 3 2 231 1 32 2 33 3 3 31 1 2 2 3 3n nn nn nm m m mn n ma x a x a x a x ba x a x a x a x ba x a x a x a x ba x a x a x a x b+ + + + =+ + + + =+ + + + =+ + + + = A solution of a system of linear equations is a sequence of numbers s1, s2,,sn that can solve each linear equation in the system 1.7 Notes: Every system of linear equations has either (1) exactly one solution (2) infinitely many solutions (3) no solution Consistent: A system of linear equations has at least one solution (for cases (1) and (2)) Inconsistent: A system of linear equations has no solution (for case (3)) 1.8 Ex 3: Solution of a system of linear equations in 2 variables (1) (2) (3) 13 = = +y xy x6 2 23= += +y xy x13= += +y xy xexactly one solutioninifinite numberno solutionlines ng intersecti twolines coincident twolines parallel two1.9 Ex 4: Using back substitution to solve a system in row-echelon form (2)(1) 25 2 == yy x Sol:By substitutinginto Eq. (1), you obtain2 = y2( 2) 5 1x x ==The system has exactly one solution:2 , 1 = = y x The row-echelon form means that it follows a stair-step pattern (two variables are in Eq. (1) and the first variable x is the leading variable, and one variable is in Eq. (2) and the second variable y is the leading variable) and has leading coefficients of 1 for all equations The back substitution means you solve Eq. (2) for y first, and then substitute the solution of y into Eq. (1) such that there is only one unknown variable x in it. Finally, you can solve x directly via Eq. (1). 1.10 Ex 5: Using back substitution to solve a system in row-echelon form (3)(2)(1)

25 39 3 2== += + zz yz y x Sol:Substituteinto (2), y can be solved as follows 2 = z 15 ) 2 ( 3 == +yyand substitute and into (1)1 = y 2 = z19 ) 2 ( 3 ) 1 ( 2== + xxThe system has exactly one solution: 2 , 1 , 1 = = = z y x1.11 Equivalent: Two systems of linear equations are called equivalent if they have precisely the same solution set Notes: Each of the following operations on a system of linear equations produces an equivalent system O1: Interchange two equations O2: Multiply an equation by a nonzero constant O3: Add a multiple of an equation to another equation Gaussian elimination: A procedure to rewrite a system of linear equations to be in row-echelon form by using the above three operations 1.12 Ex 6: Solve a system of linear equations (consistent system) (3)(2)(1) 17 5 5 24 39 3 2= + = + = + z y xy xz y xSol: First, eliminate the x-terms in Eqs. (2) and (3) based on Eq. (1) (1) (2) (2) (by O3) 2 3 93 5 (4)2 5 5 17x y zy zx y z+ + =+ = + =(1) ( 2) (3) (3) (by O3) 2 3 93 5 1 (5)x y zy zy z + + =+ = = 1.13 Since the system of linear equations is expressed in its row-echelon form, the solution can be derived by the back substitution:(only one solution) 2 , 1 , 1 = = = z y x(4) (5) (5) (by O3) 2 3 93 5 2 4 (6)x y zy zz+ + =+ ==12(6) (6) (by O2) 2 3 93 52x y zy zz + =+ ==Second, eliminate the y-terms in Eq. (5) based on Eq. (4) 1.14 Ex 7: Solve a system of linear equations (inconsistent system) (3)(2)(1)

1 3 22 2 21 33 2 13 2 13 2 1 = += = + x x xx x xx x x Sol: 1 2 32 32 3(1) ( 2) (2) (2) (by O3)(1) ( 1) (3) (3) (by O3) 3 15 4 0 (4)5 4 2 (5)x x xx xx x + + + = = = 1.15 1 2 32 3(4) ( 1) (5) (5) (by O3)3 15 4 0 0 2x x xx x + + = == So the system has no solution (an inconsistent system)(a false statement)1.16 Ex 8: Solve a system of linear equations (infinitely many solutions) (3)(2)(1) 1 31 302 13 13 2= + = = x xx xx x Sol: 1 32 31 2(1) (2) (by O1)3 1 (1)0(2)3 1 (3)x xx xx x = = + =1 32 32 3(1) (3) (3) (by O3)3 10 3 3 0 (4)x xx xx x+ = = =Let , then1.17 1 32 33 1

0x xx x = =1233 1 x tx t t Rx t= = e=t x =3,3 2x x = 3 13 1 x x + =So this system has infinitely many solutions. Since Equation (4) is the same as Equation (2), it is not necessary and can be omitted 1.18 (4) For a square matrix, the entries a11, a22, , ann are called the main diagonal entries 1.2 Gaussian Elimination and Gauss-Jordan Elimination mn matrix: (((((

mn m m mnnna a a aa a a aa a a aa a a a3 2 13 33 32 312 23 22 211 13 12 11(3) If , then the matrix is called square matrix of order nn m= Notes: (1) Every entry aij in the matrix is a real number (2) A matrix with m rows and n columns is said to be with size mn rows mcoulmns n1.19 Ex 1: Matrix Size ] 2 [((

0 00 0((

210 3 1(((

4 72 2t e1 12 24 12 3 Note: One very common use of matrices is to represent a system of linear equations (see the next slide) So a scalar (any real number) can be treated as a 11 matrix. As a result, all rules or theorems for matrices can be applied to real numbers, but not vice versa. It is not necessary to remember all rules or theorems for matrices, I suggest you to memorize those that are different for matrices and real numbers. 1.20 A system of m equations in n variables: 11 1 12 2 13 3 1 121 1 22 2 23 3 2 231 1 32 2 33 3 3 31 1 2 2 3 3n nn nn nm m m mn n ma x a x a x a x ba x a x a x a x ba x a x a x a x ba x a x a x a x b+ + + + =+ + + + =+ + + + =+ + + + =(((((

=mn m m mnnna a a aa a a aa a a aa a a aA3 2 13 33 32 312 23 22 211 13 12 11 12mbbb ( ( (= ( ( b12nxxx ( ( (= ( ( xA = x b Matrix form: 1.21 Augmented matrix: formed by concatenating the coefficient matrix with the constant-term vector of a system of linear equations 11 12 13 1 121 22 23 2 231 32 33 3 31 2 3[ ]nnnm m m mn ma a a a ba a a a ba a a a b Aa a a a b ( ( ( ( = ( ( ( b11 12 13 121 22 23 231 32 33 31 2 3nnnm m m mna a a aa a a aa a a a Aa a a a ( ( ( ( = ( ( ( Coefficient matrix: 1.22 Elementary row operations: Row equivalent: Two matrices are said to be row equivalent if one can be obtained from the other by a finite sequence of above elementary row operations (1) Interchange two rows: ( )( )ki i iM kR R (2) Multiply a row by a nonzero constant: ( ),( )ki j i j jA kR R R + (3) Add a multiple of a row to another row: , i j i jI R R Before introducing the Gaussian elimination and the Gauss-Jordan elimination, we need some background knowledge, including elementary row operations, the row-echelon form, and the reduced row-echelon form1.23 Ex 2: Elementary row operations (((

1 4 3 24 3 1 03 0 2 1(((

1 4 3 23 0 2 14 3 1 0(((

2 1 2 50 3 3 11 3 2 1(((

2 1 2 50 3 3 12 6 4 2(((

8 13 3 01 2 3 03 4 2 1(((

2 5 1 21 2 3 03 4 2 11, 2I1( )21M( 2)1,3A 1.24 Row-echelon form: (1), (2), and (3) (1) All rows consisting entirely of zeros occur at the bottom of the matrix.(2) (2) For each row that does not consist entirely of zeros, the first nonzero entry from the left side is 1, which is called as leading 1.(3) (3) For two successive nonzero rows, the leading 1 in the higher row is further to the left than the leading 1 in the lower row.(4) (4) Every column that contains a leading 1 has zeros everywhere else. Reduced row-echelon form: (1), (2), (3), and (4) 1.25 (reduced row-echelon form)(row-echelon form) Ex 3: Row-echelon form or reduced row-echelon form (((

2 1 0 03 0 1 04 1 2 1((((

1 0 0 0 04 1 0 0 02 3 1 0 03 1 2 5 1(((

0 0 0 03 1 0 05 0 1 0((((

0 0 0 03 1 0 02 0 1 01 0 0 1(((

3 1 0 01 1 2 04 3 2 1(((

4 2 1 00 0 0 02 1 2 1(row-echelon form)(reduced row-echelon form)Violate the second condition Violate the first condition 1.26 Gaussian elimination: The procedure for reducing a matrix to a row-echelon form Gauss-Jordan elimination: The procedure for reducing a matrix to its reduced row-echelon form Notes: (1) Every matrix has an unique reduced row-echelon form (2) A row-echelon form of a given matrix is not unique. (Different sequences of row operations can producedifferent row-echelon forms.) 1.27 Ex 4:Solve a system by Gauss-Jordan elimination method (only one solution) 17 5 5 24 39 3 2= + = + = + z y xy xz y x Sol: Form the augmented matrix by concatenating andA b1 2 3 91 3 0 42 5 5 17 ( ( ( ( 1 2 3 90 1 3 50 1 1 1 ( ( ( ( (1) ( 2)1,2 1,3A A 1 2 3 90 1 3 50 0 1 2 ( ( ( ( (1) (1/ 2)2,3 3A M1 0 0 10 1 0 10 0 1 2 ( ( ( ( ( 3) ( 3) (2)3,2 3,1 2,1 A A A 211= ==zyxform) echelon- (rowform) echelon- row (reduced1.28 Ex 5Solve a system by Gauss-Jordan elimination method (infinitely many solutions) 15 30 2 4 22 13 2 1= += +x xx x xSol: 1( )( 3)21,21( 2)( 1)2,122 4 2 0 1 2 1 0 1 2 1 03 5 0 1 3 5 0 1 0 1 3 11 2 1 0 1 0 5 20 1 3 1 0 1 3 1AMAM ((( ((( (( (( For the augmented matrix,(reducedrow-echelon form)(row-echelon form)1.29 Then the system of linear equations becomes1 32 3 5 23 1x xx x+ = = 3 23 13 15 2x xx x+ = =It can be further reduced toLet , thent x =3,, 3 1, 5 2321t xR t t xt x=e + = =So this system has infinitely many solutions 1.30 Homogeneous systems of linear equations : A system of linear equations is said to be homogeneous if all the constant terms are zero 00003 3 2 2 1 13 3 33 2 32 1 312 3 23 2 22 1 211 3 13 2 12 1 11= + + + += + + + += + + + += + + + +n mn m m mn nn nn nx a x a x a x ax a x a x a x ax a x a x a x ax a x a x a x a1.31 Trivial (obvious) solution: Nontrivial solution: other solutions 03 2 1= = = = =nx x x x Theorem 1.1 (1) Every homogeneous system of linear equations is consistent. Furthermore, for a homogeneous system, exactly one of the following is true: (a) The system has only the trivial solution (b) The system has infinitely many nontrivial solutions in addition to the trivial solution (In other words, if a system has any nontrivial solution, this system must have infinitely many nontrivial solutions.) (2) If the homogenous system has fewer equations than variables, then it must have an infinite number of solutions

1.32 Ex 6: Solve the following homogeneous system. Prove the above two statements numerically. 1 2 31 2 33 02 3 0x x xx x x + =+ + =((

0 1 1 00 2 0 113( )(2) (1)1,2 2 2,1A M A Let, thent x =3R t t x t x t x e = = = , , , 23 2 1solution) (trivial 0 , 0 When 3 2 1= = = = x x x tSol: ((

0 3 1 20 3 1 1For the augmented matrix,(reducedrow-echelon form)1 32 32 00x xx x+ = =